Formulation of general discrete models of thermomechanical behavior of materials with memory

Abstract

This paper is concerned with the development of general discrete models capable of depicting quite general thermomechanical behavior of a broad class of nonlinear materials with memory. Generalizations of the finite-element concept are used in conjunction with Coleman's thermodynamics of simple materials to obtain equations of motion and heat conduction for finite elements of nonlinear continua. The kinematics of finite elements is developed in general terms, with particular emphasis given to the idea that locally homogeneous deformations and temperature fields are equivalent to simplex approximations over an element. Certain basic equations of Coleman's thermodynamical theory of materials are reviewed and used to develop equations governing the behavior of a typical finite element, no restrictions being placed on the order of magnitude of the deformation gradients or temperature gradients. Topological properties of a collection of such elements are introduced to construct consistent discrete models of dissipative media with arbitrary geometry, and initial and boundary conditions.